This chapter sketches out quantum mechanics with the added purpose of clarifying further notation and adopted conventions. It presents a with a 'pragmatic' introduction to the theory, and then refers to a hypothetical "user" of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), is the subject of the second half of the chapter. Quantum entanglement, which the founding fathers of the theory used to discuss as a "spooky action at distance", has by now acquired a well-established role as a resource for quantum communication, computation and metrology. The chapter also reviews very basic results in this area. A positive but not completely positive map that serves a key role in the characterisation of quantum entanglement is transposition, in any given basis.
This chapter sketches out quantum mechanics with the added purpose of clarifying further notation and adopted conventions. It presents a with a 'pragmatic' introduction to the theory, and then refers to a hypothetical "user" of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), is the subject of the second half of the chapter. Quantum entanglement, which the founding fathers of the theory used to discuss as a "spooky action at distance", has by now acquired a well-established role as a resource for quantum communication, computation and metrology. The chapter also reviews very basic results in this area. A positive but not completely positive map that serves a key role in the characterisation of quantum entanglement is transposition, in any given basis.
This book is not meant to contain a detailed analysis of the foundations of quantum mechanics. However, we felt an introduction to the general theory was required in order to make the presentation self-sufficient and clear the following chapters from possible ambiguities. We shall, wherever possible, avoid the intricacies of the mathematical description of infinite-dimensional spaces, as well as the conceptual subtleties related to the interpretation of the formalism (although the approach taken will necessarily betray aspects of the author's stand on these matters), and content ourselves with a ‘pragmatic’ introduction to the theory, as the chapter's title suggests. Throughout the chapter, we will even occasionally refer to a hypothetical ‘user’ of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), will be the subject of the second half of the chapter, as they are very relevant to the material covered in the book.
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
This thesis establishes a number of connections between foundational issues in quantum theory, and some quantum information applications. It starts with a review of quantum contextuality and non-locality, multipartite entanglement characterisation, and of a few quantum information protocols. Quantum non-locality and contextuality are shown to be essential for different implementations of quantum information protocols known as quantum random access codes and quantum communication complexity protocols. I derive sufficient experimental conditions for tests of these quantum properties. I also discuss how the distribution of quantum information through quantum cloning processes can be useful in quantum computing. Regarding entanglement characterisation, some results are obtained relating two problems, that of additivity of the relative entropy of entanglement, and that of identifying different types of tripartite entanglement in the asymptotic regime of manipulations of many copies of a given state. The thesis ends with a description of an information processing task in which a single qubit substitutes for an arbitrarily large amount of classical communication. This result is interpreted in different ways: as a gap between quantum and classical computation space complexity; as a bound on the amount of classical communication necessary to simulate entanglement; and as a basic result on hidden-variable theories for quantum mechanics. I also show that the advantage of quantum over classical communication can be established in a feasible experiment.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
After going through the chapter on entanglement you will have reached the end of our book. As the subtitle suggests, its aim was literally to present an introduction to the subject of quantum entanglement. We have left untouched several important aspects of quantum entanglement, including multipartite systems, infinite-dimensional systems and continuous variables. Moreover, we believe that some key ideas presented in the book might be extended much further then we have managed to do. For instance the maps–states duality, illustrated in Chapter 11, might be used to find relations between capacities of quantum channels and measures of entanglement of the corresponding states of an extended system. In the book we have consistently used a geometric approach to highlight similarities and differences between the classical and quantum spaces of states. What is the knowledge gained by studying the book good for? We hope it will contribute to a better understanding of quantum mechanics. We hope also that it will provide a solid foundation for a new, emerging field of science – the theory of quantum information processing. Quantum entanglement plays a decisive role in all branches of the field including quantum cryptography, quantum error correction and quantum computing. In trying to describe the intricate geometry of the space of quantum states, we have deliberately restricted ourselves to discussing the statics of quantum theory. We have presented the arena, in which quantum information can be processed. We have not attempted to inject any concrete dynamics into our arena, but hope that readers equipped with some knowledge of its properties may introduce into it spectacular action.